Weakly holomorphic modular forms and rank two hyperbolic Kac-Moody algebras

TL;DR

This paper constructs automorphic corrections for rank 2 hyperbolic Kac-Moody algebras using weakly holomorphic modular forms and Hilbert modular forms, analyzing their Fourier coefficients and root multiplicities.

Contribution

It explicitly constructs automorphic corrections for H(a) with a=4,5,6 using Borcherds lifts, extending previous work for other values of a.

Findings

Explicit Hilbert modular forms constructed via Borcherds lifts.

Fourier coefficient asymptotics related to root multiplicities.

Automorphic correction established for new algebra cases.

Abstract

In this paper, we compute basis elements of certain spaces of weight 0 weakly holomorphic modular forms and consider the integrality of Fourier coefficients of the modular forms. We use the results to construct automorphic correction of the rank 2 hyperbolic Kac-Moody algebras H(a), a=4,5,6, through Hilbert modular forms explicitly given by Borcherds lifts of the weakly holomorphic modular forms. We also compute asymptotic of the Fourier coefficients as they are related to root multiplicities of the rank 2 hyperbolic Kac-Moody algebras. This work is a continuation of the previous paper, where automorphic correction was constructed for H(a), a=3, 11, 66.